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Theorem mulsrpr 9456
Description: Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
Assertion
Ref Expression
mulsrpr  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( [ <. A ,  B >. ]  ~R  .R  [ <. C ,  D >. ]  ~R  )  =  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )

Proof of Theorem mulsrpr
Dummy variables  x  y  z  w  v  u  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opelxpi 5021 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  -> 
<. A ,  B >.  e.  ( P.  X.  P. ) )
2 enrex 9447 . . . . 5  |-  ~R  e.  _V
32ecelqsi 7369 . . . 4  |-  ( <. A ,  B >.  e.  ( P.  X.  P. )  ->  [ <. A ,  B >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )
)
41, 3syl 16 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  [ <. A ,  B >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) )
5 opelxpi 5021 . . . 4  |-  ( ( C  e.  P.  /\  D  e.  P. )  -> 
<. C ,  D >.  e.  ( P.  X.  P. ) )
62ecelqsi 7369 . . . 4  |-  ( <. C ,  D >.  e.  ( P.  X.  P. )  ->  [ <. C ,  D >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )
)
75, 6syl 16 . . 3  |-  ( ( C  e.  P.  /\  D  e.  P. )  ->  [ <. C ,  D >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) )
84, 7anim12i 566 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( [ <. A ,  B >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  [ <. C ,  D >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) ) )
9 eqid 2443 . . . 4  |-  [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R
10 eqid 2443 . . . 4  |-  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R
119, 10pm3.2i 455 . . 3  |-  ( [
<. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )
12 eqid 2443 . . 3  |-  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  =  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R
13 opeq12 4204 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. w ,  v >.  =  <. A ,  B >. )
1413eceq1d 7350 . . . . . . . 8  |-  ( ( w  =  A  /\  v  =  B )  ->  [ <. w ,  v
>. ]  ~R  =  [ <. A ,  B >. ]  ~R  )
1514eqeq2d 2457 . . . . . . 7  |-  ( ( w  =  A  /\  v  =  B )  ->  ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  <->  [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  ) )
1615anbi1d 704 . . . . . 6  |-  ( ( w  =  A  /\  v  =  B )  ->  ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  ) 
<->  ( [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  ) ) )
17 simpl 457 . . . . . . . . . . 11  |-  ( ( w  =  A  /\  v  =  B )  ->  w  =  A )
1817oveq1d 6296 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  ( w  .P.  C
)  =  ( A  .P.  C ) )
19 simpr 461 . . . . . . . . . . 11  |-  ( ( w  =  A  /\  v  =  B )  ->  v  =  B )
2019oveq1d 6296 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  ( v  .P.  D
)  =  ( B  .P.  D ) )
2118, 20oveq12d 6299 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  ->  ( ( w  .P.  C )  +P.  ( v  .P.  D ) )  =  ( ( A  .P.  C )  +P.  ( B  .P.  D
) ) )
2217oveq1d 6296 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  ( w  .P.  D
)  =  ( A  .P.  D ) )
2319oveq1d 6296 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  ( v  .P.  C
)  =  ( B  .P.  C ) )
2422, 23oveq12d 6299 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  ->  ( ( w  .P.  D )  +P.  ( v  .P.  C ) )  =  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) )
2521, 24opeq12d 4210 . . . . . . . 8  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. ( ( w  .P.  C )  +P.  ( v  .P.  D ) ) ,  ( ( w  .P.  D )  +P.  ( v  .P.  C
) ) >.  =  <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. )
2625eceq1d 7350 . . . . . . 7  |-  ( ( w  =  A  /\  v  =  B )  ->  [ <. ( ( w  .P.  C )  +P.  ( v  .P.  D
) ) ,  ( ( w  .P.  D
)  +P.  ( v  .P.  C ) ) >. ]  ~R  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )
2726eqeq2d 2457 . . . . . 6  |-  ( ( w  =  A  /\  v  =  B )  ->  ( [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  =  [ <. ( ( w  .P.  C )  +P.  ( v  .P.  D ) ) ,  ( ( w  .P.  D )  +P.  ( v  .P.  C
) ) >. ]  ~R  <->  [
<. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D
) ) ,  ( ( A  .P.  D
)  +P.  ( B  .P.  C ) ) >. ]  ~R  ) )
2816, 27anbi12d 710 . . . . 5  |-  ( ( w  =  A  /\  v  =  B )  ->  ( ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  C )  +P.  ( v  .P.  D
) ) ,  ( ( w  .P.  D
)  +P.  ( v  .P.  C ) ) >. ]  ~R  )  <->  ( ( [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [
<. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D
) ) ,  ( ( A  .P.  D
)  +P.  ( B  .P.  C ) ) >. ]  ~R  ) ) )
2928spc2egv 3182 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( ( [
<. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D
) ) ,  ( ( A  .P.  D
)  +P.  ( B  .P.  C ) ) >. ]  ~R  )  ->  E. w E. v ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  C )  +P.  ( v  .P.  D
) ) ,  ( ( w  .P.  D
)  +P.  ( v  .P.  C ) ) >. ]  ~R  ) ) )
30 opeq12 4204 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  -> 
<. u ,  t >.  =  <. C ,  D >. )
3130eceq1d 7350 . . . . . . . . 9  |-  ( ( u  =  C  /\  t  =  D )  ->  [ <. u ,  t
>. ]  ~R  =  [ <. C ,  D >. ]  ~R  )
3231eqeq2d 2457 . . . . . . . 8  |-  ( ( u  =  C  /\  t  =  D )  ->  ( [ <. C ,  D >. ]  ~R  =  [ <. u ,  t
>. ]  ~R  <->  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  ) )
3332anbi2d 703 . . . . . . 7  |-  ( ( u  =  C  /\  t  =  D )  ->  ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  ) 
<->  ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  ) ) )
34 simpl 457 . . . . . . . . . . . 12  |-  ( ( u  =  C  /\  t  =  D )  ->  u  =  C )
3534oveq2d 6297 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  ( w  .P.  u
)  =  ( w  .P.  C ) )
36 simpr 461 . . . . . . . . . . . 12  |-  ( ( u  =  C  /\  t  =  D )  ->  t  =  D )
3736oveq2d 6297 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  ( v  .P.  t
)  =  ( v  .P.  D ) )
3835, 37oveq12d 6299 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  ->  ( ( w  .P.  u )  +P.  (
v  .P.  t )
)  =  ( ( w  .P.  C )  +P.  ( v  .P. 
D ) ) )
3936oveq2d 6297 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  ( w  .P.  t
)  =  ( w  .P.  D ) )
4034oveq2d 6297 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  ( v  .P.  u
)  =  ( v  .P.  C ) )
4139, 40oveq12d 6299 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  ->  ( ( w  .P.  t )  +P.  (
v  .P.  u )
)  =  ( ( w  .P.  D )  +P.  ( v  .P. 
C ) ) )
4238, 41opeq12d 4210 . . . . . . . . 9  |-  ( ( u  =  C  /\  t  =  D )  -> 
<. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >.  =  <. ( ( w  .P.  C )  +P.  ( v  .P.  D
) ) ,  ( ( w  .P.  D
)  +P.  ( v  .P.  C ) ) >.
)
4342eceq1d 7350 . . . . . . . 8  |-  ( ( u  =  C  /\  t  =  D )  ->  [ <. ( ( w  .P.  u )  +P.  ( v  .P.  t
) ) ,  ( ( w  .P.  t
)  +P.  ( v  .P.  u ) ) >. ]  ~R  =  [ <. ( ( w  .P.  C
)  +P.  ( v  .P.  D ) ) ,  ( ( w  .P.  D )  +P.  ( v  .P.  C ) )
>. ]  ~R  )
4443eqeq2d 2457 . . . . . . 7  |-  ( ( u  =  C  /\  t  =  D )  ->  ( [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  <->  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D
) ) ,  ( ( A  .P.  D
)  +P.  ( B  .P.  C ) ) >. ]  ~R  =  [ <. ( ( w  .P.  C
)  +P.  ( v  .P.  D ) ) ,  ( ( w  .P.  D )  +P.  ( v  .P.  C ) )
>. ]  ~R  ) )
4533, 44anbi12d 710 . . . . . 6  |-  ( ( u  =  C  /\  t  =  D )  ->  ( ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  ( v  .P.  t
) ) ,  ( ( w  .P.  t
)  +P.  ( v  .P.  u ) ) >. ]  ~R  )  <->  ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [
<. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  C )  +P.  ( v  .P.  D
) ) ,  ( ( w  .P.  D
)  +P.  ( v  .P.  C ) ) >. ]  ~R  ) ) )
4645spc2egv 3182 . . . . 5  |-  ( ( C  e.  P.  /\  D  e.  P. )  ->  ( ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  C )  +P.  ( v  .P.  D
) ) ,  ( ( w  .P.  D
)  +P.  ( v  .P.  C ) ) >. ]  ~R  )  ->  E. u E. t ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  ( v  .P.  t
) ) ,  ( ( w  .P.  t
)  +P.  ( v  .P.  u ) ) >. ]  ~R  ) ) )
47462eximdv 1699 . . . 4  |-  ( ( C  e.  P.  /\  D  e.  P. )  ->  ( E. w E. v ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  C )  +P.  ( v  .P.  D
) ) ,  ( ( w  .P.  D
)  +P.  ( v  .P.  C ) ) >. ]  ~R  )  ->  E. w E. v E. u E. t ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  ( v  .P.  t
) ) ,  ( ( w  .P.  t
)  +P.  ( v  .P.  u ) ) >. ]  ~R  ) ) )
4829, 47sylan9 657 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( (
( [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  =  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  )  ->  E. w E. v E. u E. t ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  ) ) )
4911, 12, 48mp2ani 678 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  E. w E. v E. u E. t ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  ( v  .P.  t
) ) ,  ( ( w  .P.  t
)  +P.  ( v  .P.  u ) ) >. ]  ~R  ) )
50 ecexg 7317 . . . 4  |-  (  ~R  e.  _V  ->  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  e.  _V )
512, 50ax-mp 5 . . 3  |-  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  e.  _V
52 simp1 997 . . . . . . . 8  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )  ->  x  =  [ <. A ,  B >. ]  ~R  )
5352eqeq1d 2445 . . . . . . 7  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )  -> 
( x  =  [ <. w ,  v >. ]  ~R  <->  [ <. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  ) )
54 simp2 998 . . . . . . . 8  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )  -> 
y  =  [ <. C ,  D >. ]  ~R  )
5554eqeq1d 2445 . . . . . . 7  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )  -> 
( y  =  [ <. u ,  t >. ]  ~R  <->  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  ) )
5653, 55anbi12d 710 . . . . . 6  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )  -> 
( ( x  =  [ <. w ,  v
>. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  <->  ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  ) ) )
57 simp3 999 . . . . . . 7  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )  -> 
z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )
5857eqeq1d 2445 . . . . . 6  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )  -> 
( z  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  <->  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D
) ) ,  ( ( A  .P.  D
)  +P.  ( B  .P.  C ) ) >. ]  ~R  =  [ <. ( ( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
)
5956, 58anbi12d 710 . . . . 5  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )  -> 
( ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  t >. ]  ~R  )  /\  z  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  )  <->  ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t
>. ]  ~R  )  /\  [
<. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  ( v  .P.  t
) ) ,  ( ( w  .P.  t
)  +P.  ( v  .P.  u ) ) >. ]  ~R  ) ) )
60594exbidv 1705 . . . 4  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )  -> 
( E. w E. v E. u E. t
( ( x  =  [ <. w ,  v
>. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  <->  E. w E. v E. u E. t ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  ) ) )
61 mulsrmo 9454 . . . 4  |-  ( ( x  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  y  e.  ( ( P.  X.  P. ) /.  ~R  )
)  ->  E* z E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
)
62 df-mr 9439 . . . . 5  |-  .R  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e. 
R.  /\  y  e.  R. )  /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  t >. ]  ~R  )  /\  z  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  ) ) }
63 df-nr 9437 . . . . . . . . 9  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
6463eleq2i 2521 . . . . . . . 8  |-  ( x  e.  R.  <->  x  e.  ( ( P.  X.  P. ) /.  ~R  )
)
6563eleq2i 2521 . . . . . . . 8  |-  ( y  e.  R.  <->  y  e.  ( ( P.  X.  P. ) /.  ~R  )
)
6664, 65anbi12i 697 . . . . . . 7  |-  ( ( x  e.  R.  /\  y  e.  R. )  <->  ( x  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  y  e.  ( ( P.  X.  P. ) /.  ~R  )
) )
6766anbi1i 695 . . . . . 6  |-  ( ( ( x  e.  R.  /\  y  e.  R. )  /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
)  <->  ( ( x  e.  ( ( P. 
X.  P. ) /.  ~R  )  /\  y  e.  ( ( P.  X.  P. ) /.  ~R  ) )  /\  E. w E. v E. u E. t
( ( x  =  [ <. w ,  v
>. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
) )
6867oprabbii 6337 . . . . 5  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  R.  /\  y  e.  R. )  /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  y  e.  (
( P.  X.  P. ) /.  ~R  ) )  /\  E. w E. v E. u E. t
( ( x  =  [ <. w ,  v
>. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
) }
6962, 68eqtri 2472 . . . 4  |-  .R  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  y  e.  (
( P.  X.  P. ) /.  ~R  ) )  /\  E. w E. v E. u E. t
( ( x  =  [ <. w ,  v
>. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
) }
7060, 61, 69ovig 6409 . . 3  |-  ( ( [ <. A ,  B >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  [
<. C ,  D >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  e.  _V )  ->  ( E. w E. v E. u E. t ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  )  ->  ( [ <. A ,  B >. ]  ~R  .R  [ <. C ,  D >. ]  ~R  )  =  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  ) )
7151, 70mp3an3 1314 . 2  |-  ( ( [ <. A ,  B >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  [
<. C ,  D >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) )  -> 
( E. w E. v E. u E. t
( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  )  ->  ( [ <. A ,  B >. ]  ~R  .R  [ <. C ,  D >. ]  ~R  )  =  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  ) )
728, 49, 71sylc 60 1  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( [ <. A ,  B >. ]  ~R  .R  [ <. C ,  D >. ]  ~R  )  =  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383   E.wex 1599    e. wcel 1804   _Vcvv 3095   <.cop 4020    X. cxp 4987  (class class class)co 6281   {coprab 6282   [cec 7311   /.cqs 7312   P.cnp 9240    +P. cpp 9242    .P. cmp 9243    ~R cer 9245   R.cnr 9246    .R cmr 9251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-omul 7137  df-er 7313  df-ec 7315  df-qs 7319  df-ni 9253  df-pli 9254  df-mi 9255  df-lti 9256  df-plpq 9289  df-mpq 9290  df-ltpq 9291  df-enq 9292  df-nq 9293  df-erq 9294  df-plq 9295  df-mq 9296  df-1nq 9297  df-rq 9298  df-ltnq 9299  df-np 9362  df-plp 9364  df-mp 9365  df-ltp 9366  df-enr 9436  df-nr 9437  df-mr 9439
This theorem is referenced by:  mulclsr  9464  mulcomsr  9469  mulasssr  9470  distrsr  9471  m1m1sr  9473  1idsr  9478  00sr  9479  recexsrlem  9483  mulgt0sr  9485
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